Complex symmetric Toeplitz operators on the generalized derivative Hardy space

The generalized derivative Hardy space Sα,β2(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S^{2}_{\alpha ,\beta}(\mathbb{D})$\end{document} consists of all functions whose derivatives are in the Hardy and Bergman spaces as follows: for positive integers α, β, Sα,β2(D)={f∈H(D):∥f∥Sα,β22=∥f∥H22+α+βαβ∥f′∥A22+1αβ∥f′∥H22<∞},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S^{2}_{\alpha ,\beta}(\mathbb{D})= \biggl\{ f\in H(\mathbb{D}) : \Vert {f} \Vert ^{2}_{S^{2}_{ \alpha ,\beta}}= \Vert {f} \Vert ^{2}_{H^{2}}+{ \frac{{\alpha +\beta}}{\alpha \beta}} \bigl\Vert {f'} \bigr\Vert ^{2}_{A^{2}}+ \frac{1}{\alpha \beta} \bigl\Vert {f'} \bigr\Vert ^{2}_{H^{2}}< \infty \biggr\} , $$\end{document} where H(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H({\mathbb{D}})$\end{document} denotes the space of all functions analytic on the open unit disk D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{D}}$\end{document}. In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space Sα,β2(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S^{2}_{\alpha ,\beta}(\mathbb{D})$\end{document} with respect to some conjugations Cξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\xi}$\end{document}, Cμ,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\mu , \lambda}$\end{document}. Moreover, for any conjugation C, we consider the necessary and sufficient conditions for complex symmetric Toeplitz operators with the symbol φ of the form φ(z)=∑n=1∞φˆ(−n)‾z‾n+∑n=0∞φˆ(n)zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varphi (z)=\sum_{n=1}^{\infty}\overline{\hat{\varphi}(-n)} \overline{z}^{n}+\sum_{n=0}^{\infty}\hat{\varphi}(n)z^{n}$\end{document}. Next, we also study complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space Sα,β2(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S^{2}_{\alpha ,\beta}(\mathbb{D})$\end{document}.

where H(D) denotes the space of all functions analytic on the open unit disk D. In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S 2 α,β (D) with respect to some conjugations C ξ , C μ,λ . Moreover, for any conjugation C, we consider the necessary and sufficient conditions for complex symmetric Toeplitz operators with the symbol ϕ of the form ϕ(z) = ∞ n=1φ (-n)z n + ∞ n=0φ (n)z n . Next, we also study complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space S 2 α,β (D).
where f , g ∈ L 2 (D, dA) and dA is the area measure of D. The Hilbert Hardy space H 2 (D) contains all functions f analytic on D with f (z) = ∞ n=0 a n z n , where ∞ n=0 |a n | 2 < ∞.
The Bergman space A 2 (D) consists of the space of analytic functions f in L 2 (D, dA) with f (z) = ∞ n=0 a n z n , where ∞ n=0 1 n + 1 |a n | 2 < ∞.
Let L ∞ (D) be the set of all essentially bounded measurable functions in D, and let P be the orthogonal projection from L 2 (D, dA) onto S 2 α,β (D). For ϕ ∈ L ∞ (D), the Toeplitz operator T ϕ on S 2 α,β (D) is defined by Note that, for ϕ, ψ ∈ L ∞ (D), from the definition of the Toeplitz operator, for f ∈ S 2 α,β (D) and ω ∈ D. This paper is organized as follows. First, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S 2 α,β (D) with respect to some conjugations. Moreover, we also focus on complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space S 2 α,β (D).

Complex symmetric Toeplitz operators
In this section, we study complex symmetry of Toeplitz operators on the space S 2 α,β (D). For the convenience of readers, we begin with the following lemma which comes from [9]. Let N be the natural numbers and let N 0 = N ∪ {0}. Lemma 2.1 ([9]) For s, t ∈ N 0 , the following statements hold: Remark 2.2 We mentioned in [9] that there is the difference between the Hardy space H 2 (D) (or Bergman space A 2 (D)) and the generalized derivative Hardy space S 2 α,β (D). Indeed, for s, t ∈ N 0 , the inequality z t z s ≥ P(z t z s ) holds on H 2 (D) and A 2 (D). However, it holds that z t z s ≤ P z t z s on S 2 α,β (D) because of (s+α)(s+β) (s-t+α)(s-t+β) > 1.
If C is anti-linear on S 2 α,β (D) such that Ce n = δ n e n with |δ n | = 1, then the following statements hold: where First, we will show that Parseval's identity ∞ n=0 | f , Ce n | 2 = f 2 holds for every f ∈ S 2 α,β (D). Let Cf (z) = ∞ k=0 a k z k . Then a n z n , z n = (n + α)(n + β) αβ a n .
Especially, if α = 1 and β = 2 in Theorem 2.3, then we get the following result.
Let C be anti-linear on S 2 1 (D) such that Ce n = δ n e n with |δ n | = 1. Then Parseval's identity In 2016, the authors in [8] introduced the conjugation C μ,λ on the Hardy space H 2 as in (3). Remark that the space H 2 (D) has the reproducing kernel K 1 w (z) and the normalized reproducing kernel k w (z) given by respectively. Recently, the authors in [10] gave the conjugation C ξ which has the form as in (2) on the Hardy space H 2 (D). We can easily show that the following operator as in (2) is the conjugation on S 2 α,β (D).
Then C μ,λ is a conjugation on S 2 α,β (D). Now, we establish a necessary and sufficient condition for a Toeplitz operator T ϕ on S 2 α,β (D) to be complex symmetric with respect to the above conjugations.
(ii) We claim that if P denotes the orthogonal projection of L 2 onto S 2 α,β (D), then the operators C μ,λ and P commute.
As some applications of Theorem 2.6, we get the following corollaries.
Since T ϕ is complex symmetric with the conjugation C if and only if thus equation (6) gives that φ(-k)a n z k+n .
From the constant term in (7), we have Since a k is arbitrary, we haveφ(-k) = Cφ(k) for all k ∈ N 0 . Conversely, ifφ(-k) = Cφ(k) for all k ∈ N 0 , then T ϕ is complex symmetric with the conjugation C.

Complex symmetry with non-harmonic symbols
In this section, we study the complex symmetry with non-harmonic symbols. In the Hardy space H 2 (T), z n z m is equal to z m-n , but in the generalized derivative Hardy S 2 α,β (D), z n z m = z m-n since z ∈ D. The following result gives a necessary and sufficient condition for complex symmetric Toeplitz operators with non-harmonic symbols.

β (D) is complex symmetric with the conjugation C μ,λ if and only if ϕ is either
Proof Assume that n i > m i for i ∈ N and T ϕ is complex symmetric with the conjugation Since T ϕ is a complex symmetric operator with the conjugation C μ,λ , we have that and for all i ∈ N. By equations (8) and (9), we obtain s i = m i , t i = n i , and a i = b i λ n i -m i for all i ≥ 0, and so ϕ is of the form On the one hand, suppose that ϕ is of the form Then, by similar calculations, we have that Therefore, we know that , then we can show that T ϕ is complex symmetric with the conjugation C μ,λ . This completes the proof.
if and only if s i = m i , t i = n i , and a i = b i λ n i -m i . (iii) If ϕ(z) = ∞ i=0 2a i |z| 2i for a i ∈ C, then T ϕ on S 2 α,β (D) is complex symmetric with the conjugation C μ,λ .
Proof (i) If s i = m i and t i = n i in Theorem 3.1, then we obtain statement (i).
(ii) If we put n i = m i + 1 in (i), then we get statement (ii).
(iii) If s i = m i , t i = n i , and a i = b i , then we have this result.
Example 3.4 Let ϕ(z) = ∞ j=0 (a j z n j z m j + a j e iθ z m j z n j ) for some a i ∈ C and for some real θ . Then T ϕ on S 2 α,β (D) is complex symmetric with the conjugation C μ,λ .
is not complex symmetric with the conjugation C μ,λ .
Proof The proof follows from Theorem 3.1.
Thus C μ,λ T ϕ z k = T * ϕ C μ,λ z k for any k ≥ 2, and so T ϕ is not complex symmetric Toeplitz operators. Corollary 3.7 Let ϕ(z) = az n z m + bz k , where n, m, k ∈ N with n > m and a, b ∈ C with |a| = |b|. Then T ϕ on S 2 α,β (D) is never complex symmetric with the conjugation C μ,λ .
Remark 3.8 If ϕ(z) = az n z m for a ∈ C and m, n ∈ N with m = n or ϕ(z) = z 2 z + bz for b ∈ C with b = 1. By Theorem 3.1 and Corollary 3.7, T ϕ is never complex symmetric with the conjugation C μ,λ in S 2 α,β (D) and the Hardy space H 2 (T).

Conclusion
In this paper, we make characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S 2 α,β (D) with respect to the conjugations C ξ , C μ,λ as in Theorem 2.6. Moreover, in Theorem 2.9, we deduce the necessary and sufficient conditions for complex symmetric Toeplitz operators with any conjugation C. Next, for the conjugation C μ,λ , we also obtain complex symmetric Toeplitz operators with nonharmonic symbols of the form ϕ(z) = ∞ i=0 (a i z n i z m i + b i z s i z t i ) in Theorem 3.1. The results of this paper provide an answer in the generalized derivative Hardy space S 2 α,β (D) as in the question raised in [8].